Making use of the continuum model, the phonon density of states (PDOS) are calculated analytically for single-walled carbon nanotubes (SWNT). Neglecting the interaction between twisting and longitudinal stress, the phonon dispersion relations are also determined analytically. The frequency of the breathing mode is found to be inversely proportional to r. For a (10,10) SWNT, its resonance spectral wave number is found to be 170.76 cm −1 , which is consistent with experimental observation (Science 275 (1997) 187). Four acoustic modes have been analyzed and these results are found to be in good agreement with those of Saito (Phys. Rev. B 57 (1998) 4145). Our results for the PDOS agree well with experimental data (Phys. Rev. Lett. 85 (2000) 5222).